Publication date: Jun 21, 2025
This paper proposes a fractional-order model using the Atangana-Baleanu-Caputo derivative to study the co-dynamics of tuberculosis and diabetes mellitus among susceptible (S), TB-infected (I), DM-infected (D), and co-existence (C) populations. The model’s well-posedness is established via the Banach fixed-point theorem, ensuring the uniqueness and positivity of solutions. Basic reproduction numbers (R,R,R) are derived, with values exceeding unity indicating the instability of the disease-free equilibrium and progression toward endemicity. Sensitivity analysis highlights key parameters (β,β,δ,δ,δ) affecting co-existence dynamics. Numerical simulation is conducted over T=365 days (1 year) with a unit step h=1 day, using the Adams-Bashforth method to reveal that lower fractional orders α∈(0,0. 8] slow disease decay. The model is validated against real data over 90 days at α=0. 5 using logistic growth for C(t). Results underscore the effectiveness of fractional calculus in modeling chronic co-existence and guiding control strategies.
| Concepts | Keywords |
|---|---|
| Calculus | Adams–Bashforth method |
| Ct | Fixed point theorem |
| Diabetes | |
| Slow |
Semantics
| Type | Source | Name |
|---|---|---|
| disease | MESH | tuberculosis |
| pathway | KEGG | Tuberculosis |
| disease | MESH | diabetes mellitus |
| pathway | REACTOME | Reproduction |